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Mystery Angels

An interesting story hit the news yesterday about three mysterious individuals at an obscure quant fund who secretly gave away billions of dollars to fund research into diseases such as Huntington’s disease. What they are doing is admirable, not just in terms of leaving a tremendous impact on science but also the manner in which they have gone about their philanthropy, quietly and without fanfare. A legend has grown around the campus of their quant fund, which apparently hires Ph.D.s and computer programmers. One sample interview question was given as: “For any prime number larger than 3, prove that p^2-1 is always divisible by 24. In short, show that

The question is elegant and the only mathematics you will need is to factorize p^2-1.

To show divisibility by 24, we need to account for its four factors; three 2’s and one 3. We reason as follows:

p is prime, thus p must be an odd number. Hence, (p+1) and (p-1) are even, accounting for two 2’s.

Alternating even numbers are divisible by 4, thus either (p+1) or (p-1) must be divisible by 4. This accounts for the third 2. This statement is only true if (p-1) is at least 4 meaning that p must be greater than 3 and this is presumed in the question.

For any consecutive 3 natural numbers, at least one must be divisible by three. Thus, in the running sequence (p-1), p, (p+1) with p being prime, exclusively either (p-1) or (p+1) must be divisible by 3, thus accounting for the final factor.

Neat puzzle and good start to the weekend! Many thanks to Daryl Loo of Bloomberg for making us aware of this story.

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